Finite Group Databasebeta

← #14460231

Subgroups — #14460231

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5726C211yesmaximal, characteristic, complemented
2#11752718C211yesmaximal, characteristic, complemented
3#11751180C211yesmaximal, characteristic, complemented
4C3×#65372C211yesmaximal, characteristic, complemented
5#11749281C212yesmaximal, complemented
6#11749274C212yesmaximal, complemented
7#11749212C212yesmaximal, complemented
8#11749211C212yesmaximal, complemented
9C4×#6652C211yesmaximal, characteristic, complemented
10#11746998C211yesmaximal, characteristic, complemented
11#11746883C211yesmaximal, characteristic, complemented
12#10615242—31nomaximal, Hall
13C2×C5×#239C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15#91758C2211yescharacteristic
16C5×#946C2212yes—
17C5×#895—21no—
18C5×#894—22no—
19C5×D3×C4⋊C4—21no—
20C4×C5×D12C2211yescharacteristic
21C4×C5×D12C2211yescharacteristic
22C2×C3×#688C2212yes—
23C3×C4×D20C2211yescharacteristic
24#91462—22no—
25#91456C2212yes—
26#91449—21no—
27#91358—21no—
28#91338C2212yes—
29C2×#6652C2211yescharacteristic
30C2×D3×#168C2212yes—
31#91119C2212yes—
32#91119C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C3×#3967C2212yes—
35C3×C4⋊#168C2211yescharacteristic
36#90989C2211yescharacteristic
37#90985C2212yes—
38#90967C2212yes—
39#90966C2211yescharacteristic
40#90964C2211yescharacteristic
41(C2×D3)⋊#168C2212yes—
42#80451—51nomaximal, Hall
43C5×#547—31no—
44#65897—31no—
45#65527—32no—
46#65522—32no—
47#65372D311yescharacteristic, complemented
48#65370—31no—
49#65337—32no—
50#65334—32no—
51C4×#686—31no—
52D4×#168—31no—
53#65297—31no—
54C2×C3×C5×Q8C2312yes—
55C2×Dic30—21no—
56C5×#239—24no—
57C5×#239—42no—
58C2×C5×D12—22no—
59C2×C5×D12D411yescharacteristic
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3C2312yes—
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×#1201—22no—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70C4⋊Dic15—21no—
71C4⋊Dic15C2312yes—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×(C2×D3)⋊C4—22no—
75C5×(C2×D3)⋊C4—21no—
76C5×(C2×D3)⋊C4D412yes—
77C5×C4⋊Dic3D411yescharacteristic
78C5×C4⋊Dic3C2311yescharacteristic
79C5×Dic3⋊C4—22no—
80C3×#688—42no—
81C3×#685—21no—
82C3×C4⋊Dic5C2311yescharacteristic
83C2×C3×#168C2312yes—
84(C2×D3)⋊Dic5—22no—
85#6665—22no—
86Dic3×Dic5—22no—
87#6652—22no—
88D3×#168—42no—
89#5726D511yescharacteristic, complemented
90#5563—51no—
91C3×#461—51no—
92#5172—52no—
93#5171—52no—
94#5079—51no—
95C4×#213—51no—
96#4842—51no—
97#4834—51no—
98#4815—52no—
99#4804—52no—
100C2×C5×#41—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C5×#123—62no—
103C5×C22⋊Q8—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4—31no—
107C2×C5×C4⋊C4—61no—
108C2×#688C2×D312yes—
109#4084—61no—
110C2×#686—31no—
111#4076—32no—
112#4072—32no—
113C22×#168—32no—
114#4038—61no—
115#4031—62no—
116C4×D20C2×D311yescharacteristic
117#4015—31no—
118#3967—32no—
119#3967—32no—
120#3967C2×D312yes—
121#3964—31no—
122#3964—32no—
123C22⋊#168—32no—
124#3941—31no—
125#3940—32no—
126#3939—31no—
127C4⋊#168C2×D311yescharacteristic
128C4×#168—31no—
129#3303—151noSylow
130C22×C5×D3—21no—
131C22×C5×D3C2×D412yes—
132C22×C3×D5—21no—
133C3×C5×Q8—22no—
134C3×C5×Q8—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139C2×Dic15—22no—
140Dic30—81no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3C2×D412yes—
143C5×D12—22no—
144C5×D12—21no—
145C5×D12—44no—
146C5×D12#4112yes—
147C4×C5×D3—22no—
148C4×C5×D3—22no—
149C4×C5×D3—44no—
150C4×C5×D3—42no—
151C2×C3×Dic5—21no—
152C3×D20—21no—
153C3×D20—21no—
154C3×C4×D5—42no—
155#1201—82no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3×#168—22no—
159C2×#239C2×D512yes—
160C2×C3×SD16—52no—
161C3×C4⋊Q8C2×D511yescharacteristic
162C3×C4×D4—51no—
163#946C2×D512yes—
164#938—52no—
165C2×#213—51no—
166#930—101no—
167#926—101no—
168C2×D3×C8—52no—
169#895—21no—
170#894—22no—
171#892—52no—
172D3×C4⋊C4—21no—
173#885—102no—
174C4×D12C2×D511yescharacteristic
175C4×D12C2×D511yescharacteristic
176#869—51no—
177C3×C4⋊C8—51no—
178C3×#103—52no—
179#835—52no—
180#835—52no—
181#832—51no—
182#821—52no—
183(C2×D3)⋊C8—52no—
184#817—52no—
185#809—51no—
186#807—51no—
187C4×C3⋊C8—51no—
188C5×#41—64no—
189C5×#41—122no—
190C2×C5×Q8C22×D312yes—
191C2×C5×D4—31no—
192C2×C5×D4—31no—
193C2×C5×D4—62no—
194C22×C4×C5—32no—
195C22×C4×C5—62no—
196C22×C4×C5—61no—
197C2×C22⋊D5—62no—
198C2×D20C22×D311yescharacteristic
199C2×C4×D5—21no—
200C2×Dic10—61no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—31no—
204C5×C4⋊C4—31no—
205C5×C4⋊C4—62no—
206C5×C22⋊C4—32no—
207C5×C22⋊C4—62no—
208C5×C22⋊C4—61no—
209C42×C5C22×D311yescharacteristic
210C22⋊Dic5—62no—
211#688—42no—
212#686—62no—
213#685—21no—
214#685—62no—
215C4⋊Dic5—32no—
216C4⋊Dic5—61no—
217C4⋊Dic5C22×D311yescharacteristic
218C4×Dic5—62no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—122no—
222C2×#168C22×D312yes—
223#555—151no—
224#547—31no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461—51no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5C22×D411yescharacteristic
235C2×C5×D3—22no—
236C2×C5×D3—22no—
237C2×C5×D3—22no—
238C2×C5×D3—22no—
239C2×C5×D3—41no—
240C2×C5×D3—41no—
241C2×C3×D5—41no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—21no—
245C3×C4×C5—42no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—42no—
250C3×Dic5—41no—
251C5×Dic3—22no—
252C5×Dic3—41no—
253C2×C3×Q8C22×D512yes—
254C2×C3×D4—51no—
255C22×C3×C4—101no—
256C2×#67—102no—
257#239—24no—
258#239—42no—
259C2×D12—22no—
260C2×D12C22⋊D511yescharacteristic
261C2×D12C22×D511yescharacteristic
262C2×C4×D3—21no—
263C2×C4×D3—22no—
264C2×C4×D3C22×D512yes—
265C2×Dic6—101no—
266C3×SD16—202no—
267C2×C3×C8—52no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—51no—
271C3×C22⋊C4—101no—
272C3×C42C22×D511yescharacteristic
273#217—102no—
274#213—102no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—21no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4C22⋊D512yes—
279C4⋊Dic3—52no—
280C4⋊Dic3—101no—
281C4⋊Dic3C22⋊D511yescharacteristic
282C4⋊Dic3C22×D511yescharacteristic
283Dic3⋊C4—22no—
284C4×Dic3—102no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—51no—
287D3×C8—202no—
288C23×C5—32no—
289C23×C5—61no—
290C22×D5—21no—
291C5×Q8—22no—
292C5×Q8—22no—
293C5×D4—32no—
294C5×D4—62no—
295C5×D4—61no—
296C5×D4—124no—
297C2×C4×C5—22no—
298C2×C4×C5—32no—
299C2×C4×C5—62no—
300C2×C4×C5—62no—
301C2×C4×C5—61no—
302C2×C4×C5—124no—
303C2×C4×C5—122no—
304C2×C4×C5D3×D411yescharacteristic
305C2×C4×C5D3×D411yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—242no—
308C2×Dic5—21no—
309C2×Dic5—62no—
310D20—21no—
311D20—21no—
312C4×D5—42no—
313Dic10—241no—
314#168—22no—
315#168—61no—
316#168—61no—
317C2×#41—32no—
318C2×SD16—52no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—151no—
322C4⋊Q8D3×D511yescharacteristic
323#124—302no—
324#123—62no—
325C22⋊Q8—32no—
326#121—61no—
327#121—152no—
328#121—301no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—51no—
332#117—152no—
333#117—301no—
334C2×C4⋊C4—61no—
335#107—151no—
336#106—152no—
337C4⋊C8—51no—
338#103—52no—
339#103—152no—
340#103—152no—
341#102—151no—
342#102—151no—
343#102—152no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5#54611yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5#57711yescharacteristic
349C3×D5—81no—
350C5×D3—42no—
351C5×D3—42no—
352C23×C3—101no—
353C22×D3—21no—
354C22×D3C2×C22⋊D512yes—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×D4—101no—
358C3×D4—101no—
359C2×C3×C4—22no—
360C2×C3×C4—101no—
361C2×C3×C4—202no—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C2×C22⋊D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—402no—
366C2×Dic3—21no—
367C2×Dic3—102no—
368C2×Dic3C2×C22⋊D512yes—
369D12—22no—
370D12—21no—
371D12—44no—
372D12#71112yes—
373C4×D3—22no—
374C4×D3—22no—
375C4×D3—44no—
376C4×D3—42no—
377Dic6—401no—
378C3×C8—102no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—62no—
384C22×C5—62no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—41no—
389C2×D5—41no—
390C2×D5—41no—
391C4×C5—21no—
392C4×C5—42no—
393C4×C5—62no—
394C4×C5—121no—
395C4×C5#91612yes—
396C4×C5#100411yescharacteristic
397C4×C5D3×#4111yescharacteristic
398Dic5—41no—
399Dic5—122no—
400#41—64no—
401#41—122no—
402C2×Q8—301no—
403C2×Q8C2×D3×D512yes—
404C2×D4—31no—
405C2×D4—31no—
406C2×D4—51no—
407C2×D4—62no—
408C2×D4—302no—
409C22×C4—32no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—101no—
413SD16—202no—
414D8—302no—
415C2×C8—52no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—31no—
422C4⋊C4—31no—
423C4⋊C4—51no—
424C4⋊C4—62no—
425C4⋊C4—152no—
426C4⋊C4—301no—
427C22⋊C4—32no—
428C22⋊C4—62no—
429C22⋊C4—61no—
430C22⋊C4—101no—
431C22⋊C4—302no—
432C22⋊C4—302no—
433C42—302no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—201no—
437C22×C3—201no—
438C22×C3—201no—
439C22×C3C22×C22⋊D511yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—41no—
445C2×D3—41no—
446C3×C4—21no—
447C3×C4—42no—
448C3×C4—201no—
449C3×C4#409612yes—
450C3×C4C2×#71111yescharacteristic
451C3×C4#414411yescharacteristic
452Dic3—22no—
453Dic3—41no—
454Dic3—202no—
455C2×C5—122no—
456C2×C5—122no—
457C2×C5#561211yescharacteristic
458C2×C5C2×#91611yescharacteristic
459C2×C5#577811yescharacteristic
460D5—81no—
461C23—32no—
462C23—61no—
463C23—101no—
464Q8—22no—
465Q8—22no—
466Q8—1201no—
467D4—32no—
468D4—61no—
469D4—62no—
470D4—101no—
471D4—101no—
472D4—124no—
473D4—1202no—
474C2×C4—22no—
475C2×C4—32no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—101no—
480C2×C4—124no—
481C2×C4—122no—
482C2×C4—202no—
483C2×C4—302no—
484C2×C4D3×C22⋊D511yescharacteristic
485C2×C4D3×C22⋊D511yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—102no—
488C8—301no—
489C8—301no—
490C2×C3—401no—
491C2×C3#6614011yescharacteristic
492C2×C3C2×#409611yescharacteristic
493C2×C3#6617111yescharacteristic
494D3—42no—
495D3—42no—
496C5#8045111yescharacteristic, complemented, Sylow
497C22—62no—
498C22—62no—
499C22—62no—
500C22—62no—
501C22—121no—
502C22—121no—
503C22—201no—
504C22—201no—
505C22—201no—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—21no—
508C4—42no—
509C4—62no—
510C4—121no—
511C4—201no—
512C4—602no—
513C4#9151712yes—
514C4D3×#71111yescharacteristic
515C4#9202511yescharacteristic
516C3#1061524211yescharacteristic, complemented, Sylow
517C2—122no—
518C2—122no—
519C2—401no—
520C2#1175494611yescharacteristic
521C2#1175509211yescharacteristic
522C2C2×#9151711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3303
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×#5726C2semidirect
2. #11752718C2semidirect
3. #11751180C2semidirect
4. C3×#65372C2semidirect
5. #11749281C2semidirect
6. #11749274C2semidirect
7. #11749212C2semidirect
8. #11749211C2semidirect
9. C4×#6652C2semidirect
10. #11746998C2semidirect
11. #11746883C2semidirect
47. #65372D3semidirect
89. #5726D5semidirect
435. C3×C5#3303semidirect
496. C5#80451semidirect
516. C3#10615242semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#57262
#117527182
#117511802
C3×#653722
#117492812
#117492742
#117492122
#117492112
C4×#66522
#117469982
#117468832
#106152423
#804515